论文标题
通用量子门效率的可计算下限
Calculable lower bounds on the efficiency of universal sets of quantum gates
论文作者
论文摘要
当前可用的量子计算机，即所谓的嘈杂的中间尺度量子（NISQ）设备，其特征是量子数量相对较少和中等的门贴。在这种情况下，量子误差校正的实现是不可能的，这些设备的性能非常适中。特别是，具有相当高的忠诚度可实现的电路深度是有限的，并且需要最小化电路深度。这样的深度取决于计算中使用的通用门集$ \ Mathcal {s} $的效率，并且可以使用Solovay-Kitaev Theorem进行界定。但是，众所周知，可以为特定的$ \ mathcal {s} $获得$ \ mathcal {o}（\ mathrm {log}（ε^{ - 1}））$的更好，渐近紧密的界限（\ mathrm {log}（ε^{ - 1}））$。这些界限由所谓的光谱差距控制，表示为$ \ mathrm {gap}（\ Mathcal {s}）$。然而，对于一般$ \ mathcal {s} $，不可能进行$ \ mathrm {gap}（\ mathcal {s}）$的计算，而实际上，一个人认为频谱差距为$ r（ε）$，表示为$ r（ε）$，表示为$ \ mthrm {gap} _r {gap} _r _ _ _ mathcal（\ mathcal（\ nathcal）$。事实证明，这足以绑定$ \ MATHCAL {S} $的效率，前提是人们对物理上可行的情况感兴趣，在该情况下，错误$ε$从下面有限。在本文中，我们在$ \ mathrm {gap} _r（\ Mathcal {s}）$上得出了下限，并因此在$ d $ d $ d $ d $ dimensional量子$ \ Mathcal $ \ mathcal {s} $的通用效率上获得了满足附加条件的效率。对于通用量子门，例如，例如HAAR随机门。我们的界限是明确的，因为所有参数都可以通过现有计算机上的数值计算来确定，至少对于小$ d $。这与$ \ mathrm {gap} _r（\ mathcal {s}）$上的已知下限相反，该$涉及具有模棱两可的值的参数。
Currently available quantum computers, so called Noisy Intermediate-Scale Quantum (NISQ) devices, are characterized by relatively low number of qubits and moderate gate fidelities. In such scenario, the implementation of quantum error correction is impossible and the performance of those devices is quite modest. In particular, the depth of circuits implementable with reasonably high fidelity is limited, and the minimization of circuit depth is required. Such depths depend on the efficiency of the universal set of gates $\mathcal{S}$ used in computation, and can be bounded using the Solovay-Kitaev theorem. However, it is known that much better, asymptotically tight bounds of the form $\mathcal{O}(\mathrm{log}(ε^{-1}))$, can be obtained for specific $\mathcal{S}$. Those bounds are controlled by so called spectral gap, denoted $\mathrm{gap}(\mathcal{S})$. Yet, the computation of $\mathrm{gap}(\mathcal{S})$ is not possible for general $\mathcal{S}$ and in practice one considers spectral gap at a certain scale $r(ε)$, denoted $\mathrm{gap}_r(\mathcal{S})$. This turns out to be sufficient to bound the efficiency of $\mathcal{S}$ provided that one is interested in a physically feasible case, in which an error $ε$ is bounded from below. In this paper we derive lower bounds on $\mathrm{gap}_r(\mathcal{S})$ and, as a consequence, on the efficiency of universal sets of $d$-dimensional quantum gates $\mathcal{S}$ satisfying an additional condition. The condition is naturally met for generic quantum gates, such as e.g. Haar random gates. Our bounds are explicit in the sense that all parameters can be determined by numerical calculations on existing computers, at least for small $d$. This is in contrast with known lower bounds on $\mathrm{gap}_r(\mathcal{S})$ which involve parameters with ambiguous values.
